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Absolute Continuity
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Definition of Absolute Continuity of a Function
A function f is absolutely continuous on an interval [a, b] if for every there exists such that for any finite collection of disjoint sub-intervals , the sum of the lengths of the sub-intervals is less than implies the sum of is less than .
Absolute Continuity vs Uniform Continuity
A function that is absolutely continuous on an interval is also uniformly continuous on that interval, but the converse is not necessarily true.
Absolute Continuity on Compact Intervals
If a function is absolutely continuous on [a, b], then it is continuous on [a, b] and has a derivative almost everywhere on [a, b], such that the integral of over [a, b] equals .
Characterization by Derivatives
If is differentiable almost everywhere on [a, b], has a Lebesgue integrable derivative , and for all in [a, b], then is absolutely continuous on [a, b].
Absolute Continuity and Variation
A function is absolutely continuous on [a, b] if and only if it is both continuous and of bounded variation on that interval.
Relation to Almost Everywhere Differentiability
An absolutely continuous function on a compact interval is differentiable almost everywhere on that interval.
Lebesgue's Characterization of Absolute Continuity
Lebesgue's theorem states that a continuous function of bounded variation on [a, b] is absolutely continuous if and only if its set of discontinuities for its derivative is of Lebesgue measure zero.
Absolute Continuity of Measures
A measure is absolutely continuous with respect to another measure (denoted ) if implies for every measurable set E.
Radon-Nikodym Theorem
The Radon-Nikodym theorem provides that if , then there exists an integrable function such that for all measurable sets E.
Total Variation of an Absolutely Continuous Function
The total variation of an absolutely continuous function on an interval [a, b] can be computed as the integral of the absolute value of its derivative: .
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